Cooling Tower Performance Curves: Reading the Crossplot

A cooling tower is rarely rated by a single curve. The performance guarantee that a plant actually buys is a family of them: for each wet-bulb temperature and each cooling range, a curve showing the cold-water temperature the tower should produce across a range of water flows. The test then has to answer a two-dimensional question — did the tower do what this family says it should, at the flow and weather we actually have? — and the tool for that is the crossplot.

What a performance curve family contains

Each record in a performance curve set is a measured or predicted point:

{
  wet-bulb temperature,
  range,
  water flow,
  cold-water temperature
}

Hold the wet bulb and the range, and the set becomes a simple two-dimensional curve: cold-water temperature against water flow, rising as flow increases because each square metre of fill has to absorb more heat. Hold the flow and the set becomes a map across weather and load. The family is what lets an owner compare a test performed in May with a guarantee written for the hottest design day.

This is a different route to the same question the characteristic method answers. The characteristic method needs the tower’s KaV/L curve and integrates; the performance-curve method needs measured points at several flows and interpolates. Neither replaces the other: the characteristic method is compact and depends on a fill exponent, while the performance-curve method is data-hungry but makes no assumption about the shape of the fill’s behaviour.

Interpolating inside the family

Two interpolation steps are needed, and they happen in this order.

1. Across water flow, within a corner. At each of the four corners of the wet-bulb/range rectangle surrounding the test condition, the cold-water temperature is interpolated against water flow.
2. Across wet bulb and range. The four corner values are then combined bilinearly to the test’s wet bulb and range.

The order matters because cold-water temperature is far more strongly curved against flow than against wet bulb. Interpolating across the rectangle first and against flow afterwards would smooth away the curvature that carries most of the information.

The inverse solve: from temperature back to flow

A test does not arrive as “the flow the curve predicts”. It arrives as a measured cold-water temperature at a measured flow. Turning that into a capability percentage means inverting the interpolation — solving for the flow at which the curve family predicts the tested temperature:

CWTgrid(WB, range, qpredicted) − CWTtest = 0

Because the curve rises monotonically with flow over its valid span, this is a well-behaved single root: bracket it between the smallest and largest tested flow, then solve. The root is qpredicted — the flow the guarantee’s curve says would have produced exactly the cold water that was measured.

Capability is then the ratio of what the tower handled to what the curve predicted:

Capability % = 100 × qtest, adjusted / qpredicted

Leaving-water deviation, and the sign that flips

A capability percentage answers the contract question. Operators usually want the more physical one: how many degrees warmer or colder than predicted?

ΔTc = Tctest − Tcpredicted at the adjusted flow

Under the convention above, a positive deviation means the tested cold water was warmer than the curve predicts — worse than the rating. That sign convention is not universal, and a report that omits it is ambiguous: a reader who assumes the opposite convention will read a 0.3 K shortfall as a 0.3 K gain. State the convention alongside the number.

Illustrative example

The following numbers are an illustrative example, not a measured result. They are chosen to make the arithmetic legible.

Step 1 — read the curve family at the test wet bulb, range and flow:

CWTgrid(27.0, 10.0, 200) = 32.5 °C

Step 2 — deviation, tested minus predicted:

ΔTc = 32.8 − 32.5 = +0.3 K (warmer than predicted, worse)

Step 3 — invert for the flow at which the curve predicts 32.8 °C:

qpredicted = 210 kg/s

Step 4 — capability:

Capability % = 100 × 200 ÷ 210 = 95.2%

The two answers agree, which is the point of the exercise: 0.3 K of leaving-water deviation is worth about 5% of flow capability at this duty. That ratio is not a constant — it tightens near the pinch, where the curve becomes steep and a tenth of a degree is worth several percent of flow.

Before any of this: corrections

The curve and the test are only comparable once both are on the same basis. In an acceptance test that means the prescribed corrections are applied before the interpolation: to the water flow, to the fan power or airflow, and to the atmospheric condition the test was run in, along with the validity checks that decide whether the test may be evaluated at all. The correction procedure belongs to the test code and to the project’s contractual reference; it is not reproduced here, and applying a capability calculation without it produces a number that looks precise and is not.

Using a performance curve well

• Stay inside the grid. Interpolation inside the tested wet-bulb, range and flow envelope is sound. Extrapolation past the tested flow range or beyond the highest tested wet bulb invents curve shape rather than reading it.
• Mind how many flows each curve was built from. A curve fitted through two points is a straight line between them, and the inverse solve will return very confident numbers from very thin data.
• Check the range coverage. A guarantee curve set that covers a 5–8 K range will not answer a test run at a 12 K range, which is common when a plant increases production.
• Keep the corrections and the raw data in the report. The crossplot is a conclusion; the measurements that fed it are the evidence.

Read more

• Inspection and performance testing service
• Cooling tower capability explained: what 100% really means
• The Merkel number and the entering-air convention
• Cooling tower water too warm? 7 checks before a major repair
• Cooling tower glossary

Sources

• Hensley, J.C. (ed.) — Cooling Tower Fundamentals, 2nd edition (SPX Cooling Technologies, 2009) — the performance-curve family this article describes, and the two code measures of capability it reproduces — test flow over predicted flow, and the characteristic-curve alternative.
• CTI ATC-105 — Acceptance Test Code for Water Cooling Towers — the Cooling Technology Institute’s acceptance test code for water cooling towers, referenced by this article’s statement that the corrections before the interpolation belong to the test code; not reproduced here.
• Kloppers & Kröger — Cooling tower performance: a critical evaluation of the Merkel assumptions (SAIMechE, 2004) — the Merkel demand the curves of the chart are inverted against.
• ASHRAE Handbook—Fundamentals (2025 edition) — Psychrometrics — the moist-air relations the chart’s saturation-enthalpy values are computed from.
• Buck Research Instruments — CR-1A User’s Manual, Appendix 1: Humidity Conversion Equations (revised 7/96) — the saturation vapour pressure the chart evaluates.